Phy 211 General Physics I Chapter 7 Kinetic Energy Work Lecture Notes What is Energy 1 Energy is a scalar quantity associated with the state of an object or system of objects Energy is a calculated quantity that appears in nature and whose total quantity in a system always remains constant and accounted for Energy is said to be Conserved 2 Loosely speaking energy represents the fuel necessary for changes to occur in the universe and is often referred to as the capacity to perform work 3 We can think of energy as the currency associated with the transactions forces that occur in nature In mechanical systems energy is spent as force transactions are conducted Alternatively the exertion of force requires an expenditure of energy 4 The SI units for energy are called Joules J In honor of James Prescott Joule James Prescott Joule 1818 1889 1 English inventor scientist 2 Interested in the efficiency of electric motors 3 Described the heat dissipated across a resistor in electrical circuits now known as Joules Law 4 Demonstrated that heat is produced by the motion of atoms and or molecules 5 Credited with establishing the mechanical energy equivalent of heat 6 Participated in establishing the Law of Energy Conservation Energy is never created nor destroyed it is merely transformed from 1 form to another Therefore the total energy of a closed system remains constant unless affected upon by a force external to the system Review The Scalar Dot Product r r 1 Two vectors A and B can be multiplied to product a scalar resultant called the scalar or Dot product r r r r B A B cosf 2 When using the magnitudes of the vectors A where is the angle between vectors A and B r r B A x Bx A y By 3 When using vector components A 4 Useful properties of scalar r rproducts r r B B A a Commutative property r rA 2 A A A b Squaring vectors i j j k k 1 c Unit vectors i Important Application We will use the scalar product of the vectors of force and displacement to calculate work performed by force Kinetic Energy The energy associated with an object s state of motion Kinetic energy is a scalar quantity that is never negative in value 2 Definition 1 m 2 K mv SI units kg Scalar form 2 s 1 r r 1 1 1 v mv2x mv2y mv2z Vector form K mv 2 2 2 2 K Kx Ky Kz Key Notes 1 The kinetic energy for the x y and z components are additive 2 Kinetic energy is relative to the motion of observer s reference frame since speed and velocity are as well 3 An object s kinetic energy depends more on its speed than its mass 4 Any change in an object s speed will affect a change in its kinetic energy 2 m 5 Unit comparison 1 J 1 kg s Work The energy transferred to from an object by the exertion of a force 1 Work is essentially a measure of useful physical output or Work Effort x Outcome r Dr r F Definition standard form W or W r r F Dr Fx Dx Fy Dy Fz Dz r r F Dr cosf 2 SI Units for Work N m r F Notes Unit equivalence 1 J 1 N m r Dr Work cont 1 The general definition of Work for a varying as well as W r r r constant force W dW F dr 0 ro 2 Graphically work is the area beneath the Force vs Displacement graph Fcos Fcos Area Area r r r dW F dr W r r r dW F dr W r 0 ro Work Kinetic Energy 1 The net work performed to r on an r objectr is related r Dr m a Dr the net force WNet FNet r 2 When FNet 0 a change in state of motion kinetic energy is implied r r WNet FNet dr K K o DK Derivation r WNet WNet WNet WNet r r m a dr r r r r r r dv dv dr r m a dr m dr m r dr dt dr dt r r r r r v 1 r dr m dv m v dv m v v vo 2 dt r r r r 1 1 m v v m vo vo The Work Energy Theorem 2 2 r dW FNet dr Work Performed by Gravitational Force r For a falling body no air drag r r Wy Fg Dy mgDy since cosf 1 r r Wx Fg Dx 0 since cosf 0 Wg Wx Wy mgDy Gravitational force only performs work in the vertical direction 1 Wg is when y is r Fg mg r Dy r r Fg mg 2 Wg is when y is What about gravitational force on an incline r Dy Work Performed during Lifting Lowering Consider Joey blasting his pecs with a bench press workout assume vlift constant r mg Given mbar 100 kg r mbar g 980 N j Dy 1m j r Dy nd Law Applying Newton s 2 r r FNet FLift mbar g j mbar a 0 FLift mbar g 980 N The Work performed r r Wg mbar g Dy 980 N m r WLift FLift Dy 980 N m r r WNet FNet Dy WLift Wg 0 r FLift Work Performed by Elastic Force 1 The elastic or restoring force associated with the compression or stretching of a simple spring increases linearly with the amount of deformation this is Hooke s Law 2 The following assumptions are made The spring is ideal massless The spring obeys Hooke s Law r r r r r r Fspring kDrstretch kx where Drstretch Dx x r where x o 0 is the equilibrium position of the spring and k is the spring constant in N m Work Performed by Elastic Force cont Consider a spring k 10 N m attached to a wall at one end and a mass 1 kg at the other The block is free to slide on a frictionless surface The block spring system is compressed 0 5 m then released The work performed by the spring force is r Wspring Fspring dx Wspring Wspring m m m Equilibrium Position r r 1 r r x 1 kx dx k x x xo k x2 x20 xo 2 2 1 2 1 2 kx0 kx 2 2 x r Fspring r Fspring Power 1 A measure of work effectiveness 2 The time rate of energy transfer work due to an exerted force r r dW F dr P dt dt r r W F Dr Average Power Pavg Dt Dt r r dW F dr Instantaneous Power P dt dt r r DWNet FNet dr DK Average Net Power PNet Dt Dt Dt 3 SI units The Watt 1 W 1 J s Note Power is also related to Force Velocity r r r r dW F dr P F v dt dt Power cont The same work output can be performed at …